Remember that pages on a domain don't exist in isolation. Domains on a C-block don't exist in isolation. Resources are interconnected. One requires some holistic systems thinking to properly determine when it's inappropriate not to SEO a page or a domain. Nevertheless, SEO is not always appropriate right?
When is SEO inappropriate?
SEO is inappropriate when one doesn't want a page or domain to be found, crawled, or indexed by a search engine.
Motives for this isolation vary, and include:
Privacy
When you want to hide content from the general public for some reason, you have no incentive to optimize a site for search engines. You may want to keep a site private because you're just using it as long term storage of your data, or because you're saving copyrighted content there for download.
Exclusivity
If you're offering something you don't want the general public to have a super special album download or highly targeted white paper you need not necessarily optimize those pages. Although for both of the previous examples, one could gain more exposure with an optimized site, some don't intend to attract attention that way, and instead opt for "exclusive" traffic through, say, word of mouth or invitation. And it's not just traditional "content" here, but social media sites as well, who may not want to optimize their sites, as they are pursue a word-of-mouth marketing strategy with invitations and walled gardens to launch. Facebook, gmail, and oink.cn all grew this way in their early days, without intentionally optimizing anything for search engines.
Security
For example, say, you're an SEO looking to improve your domain's page load time, so you serve static content through a cookieless domain. Although the cookieless domain is used to improve the SEO of another domain, the cookieless domain need not be optimized itself for search engines; and probably shouldn't be. The security of a domain used as a CDN is a good reason not to promote it with SEO.
Testing In Isolation
Let's say you want to measure how many people link to a site within a year which is completely promoted with AdWords, and through no other medium. Hey, some people like doing that sort of thing to figure out if there's a market demand for something, or to test user experience while building a site. I may not understand the reasoning to isolate things using this reasoning, personally, but I do understand that when people want isolated tests excluding SEO, SEO would be inappropriate.
When One's Business Doesn't Rely On The Web For Traffic, Nor Would They Want To
Many local businesses or businesses which rely on point-of-sale or earning their traffic through some other mechanism than digital marketing may not want to even consider optimizing their site for search engines because they've already optimized it for some other system, like people walking down a street after emptying out of bars or an amusement park.
When Competing For Things Nobody Is Searching For
Objection! Just because someone isn't searching for the keyword now doesn't mean they won't search for it later en masse! Ok, we SEO's need to suspend our disbelief for a minute. Many business owners, developers, engineers, artists and other folk believe that people won't search for keywords associated with their work, so it's irrelevant to optimize their work for those keywords -- and, in fact, they may very well be correct. Not everything has a high search volume, and some things do better via discovery (social) than search, for instance, Animal Training Unlimited gets many referrals through their social network and from people who see their animal training skills, and not a whole lot from search. The delicious Kunfusion food truck, similarly, receives its traffic from social media, word-of-mouth, repeat customers, and, well, the geography surrounding where it parks.
When Competing Differently In An A Saturated Market
Let's say you want to market entirely through social media, and you don't have to worry much about SEO. Let's say you're Spotify (in the saturated market of streaming music), Zynga (in the saturated casual gaming market), or Pinterest (in the saturated photo sharing market). Likewise, if you're competing in a highly competitive, "saturated" SERP, you may prefer to pursue traffic through social media than through SEO -- for instance, consider the competition in the SERPs for "how to calm down" related to the destination of that link.
Any other examples? Have you ever advised your self or an employer or client that "SEO, in this case, may not be appropriate"? Do tell!